# Improved quantum lower and upper bounds for matrix scaling

Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting. We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for n × n matrices with at most m non-zero entries and with ℓ_2-error ε=Θ(1/m) must make Ω(m) queries to the matrix, even when the success probability is exponentially small in n. Additionally, we show that for ε∈[1/n,1/2], any quantum algorithm capable of producing ε/100-ℓ_1-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε) queries, and that there exists a constant ε_0>0 for which this problem takes Ω(n^1.5) queries. To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-ε regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε-ℓ_1-scaling in time O(n^1.5/ε^2), whereas the best previously known bounds were O(n^2polylog(1/ε)) (classical) and O(n^1.5/ε^3) (quantum).

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